Redistributive Allocation Mechanisms

1. Redistributive Allocation Mechanisms (with an application to the potential European energy crisis) Mohammad Akbarpour r O Piotr Dworczak r O Scott Duke Kominers* (Stanford) (GRAPE & Northwestern) (Harvard) 22 November, 2022 European Central Bank, invited speaker seminar Research sponsored by ERC grant IMD-101040122. *Authors’ names are in certified random order.

2. Motivation Many goods and services are allocated using non-market mechanisms, even though monetary transfers are feasible: Housing

3. Motivation Many goods and services are allocated using non-market mechanisms, even though monetary transfers are feasible: Housing Health care

4. Motivation Many goods and services are allocated using non-market mechanisms, even though monetary transfers are feasible: Housing Health care Food

5. Motivation Many goods and services are allocated using non-market mechanisms, even though monetary transfers are feasible: Housing Health care Food Road access

6. Motivation Many goods and services are allocated using non-market mechanisms, even though monetary transfers are feasible: Housing Health care Food Road access ...

7. Motivation Many goods and services are allocated using non-market mechanisms, even though monetary transfers are feasible: Housing Health care Food Road access ... Follow-up paper: Covid-19 vaccines

8. Motivation Many goods and services are allocated using non-market mechanisms, even though monetary transfers are feasible: Housing Health care Food Road access ... Follow-up paper: Covid-19 vaccines Focus today: Energy this winter in Europe

9. Motivation—European energy crisis Source: Sgaravatti, G., S. Tagliapietra, G. Zachmann (2021) ‘National policies to shield consumers from rising energy prices’

12. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT)

13. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152

14. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information!

15. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information! Governments do have some information... but far from perfect in practice:

16. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information! Governments do have some information... but far from perfect in practice: ▶ Detailed financial situation?

17. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information! Governments do have some information... but far from perfect in practice: ▶ Detailed financial situation? ▶ Job market opportunities?

18. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information! Governments do have some information... but far from perfect in practice: ▶ Detailed financial situation? ▶ Job market opportunities? ▶ Exposure to interest rate hikes?

19. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information! Governments do have some information... but far from perfect in practice: ▶ Detailed financial situation? ▶ Job market opportunities? ▶ Exposure to interest rate hikes? ▶ Energy efficiency of housing?

20. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information! Governments do have some information... but far from perfect in practice: ▶ Detailed financial situation? ▶ Job market opportunities? ▶ Exposure to interest rate hikes? ▶ Energy efficiency of housing? ▶ Heating method?

21. Motivation Classical economic perspective: Distorting prices will lead to inefficiency (I WT); if you want to redistribute, do so via lump-sum payments (II WT) ▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152 But: Welfare theorems ignore the issue of private information! Governments do have some information... but far from perfect in practice: ▶ Detailed financial situation? ▶ Job market opportunities? ▶ Exposure to interest rate hikes? ▶ Energy efficiency of housing? ▶ Heating method? ▶ Family size?

22. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market)

23. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market) Designer allocates a set of goods to agents differing in their observed and unobserved characteristics.

24. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market) Designer allocates a set of goods to agents differing in their observed and unobserved characteristics. We derive the optimal mechanism under IC and IR constraints and an objective that reflects redistributive concerns via welfare weights.

25. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market) Designer allocates a set of goods to agents differing in their observed and unobserved characteristics. We derive the optimal mechanism under IC and IR constraints and an objective that reflects redistributive concerns via welfare weights. Some key take-aways:

26. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market) Designer allocates a set of goods to agents differing in their observed and unobserved characteristics. We derive the optimal mechanism under IC and IR constraints and an objective that reflects redistributive concerns via welfare weights. Some key take-aways: ▶ Non-market mechanisms can be optimal!

27. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market) Designer allocates a set of goods to agents differing in their observed and unobserved characteristics. We derive the optimal mechanism under IC and IR constraints and an objective that reflects redistributive concerns via welfare weights. Some key take-aways: ▶ Non-market mechanisms can be optimal! ▶ If lump-sum payments are available, uniform price caps are never optimal;

28. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market) Designer allocates a set of goods to agents differing in their observed and unobserved characteristics. We derive the optimal mechanism under IC and IR constraints and an objective that reflects redistributive concerns via welfare weights. Some key take-aways: ▶ Non-market mechanisms can be optimal! ▶ If lump-sum payments are available, uniform price caps are never optimal; ▶ Instead, price caps should only be applied up to some threshold consumption level;

29. Inequality-aware Market Design Our approach: A market-design framework for determining the optimal allocation under redistributive concerns (Market-design perspective: the designer only controls a single market) Designer allocates a set of goods to agents differing in their observed and unobserved characteristics. We derive the optimal mechanism under IC and IR constraints and an objective that reflects redistributive concerns via welfare weights. Some key take-aways: ▶ Non-market mechanisms can be optimal! ▶ If lump-sum payments are available, uniform price caps are never optimal; ▶ Instead, price caps should only be applied up to some threshold consumption level; ▶ Uniform price caps can be optimal when lump-sum payments are costly;

30. Literature Weitzman (1977): First paper to point out that random allocation might be preferred to market pricing when agents’ “needs” are not well expressed by willingness to pay. Condorelli (2013): A full mechanism-design approach to Weitzman’s problem ▶ On a technical level: We add heterogeneous quality of objects, continuum of goods and agents, groups of agents with the same observable characteristics, and designer’s preferences over revenue. ▶ On a conceptual level: We focus on a particular objective function (weighted sum of utilities and revenue) and connect market circumstances with practical conclusions about the optimal design. Dworczak r O Kominers r O Akbarpour (2021): two-sided market but “maximally simplified” (no heterogeneous quality, no observable information, no restrictions on lump-sum transfers)

31. Literature Price control, public provision of goods: Nichols and Zeckhauser (1982), Besley and Coate (1991), Blackorby and Donaldson (1988), Viscusi, Harrington, and Vernon (2005), Gahvari and Mattos (2007), Currie and Gahvari (2008) Optimal taxation: Diamond and Mirrlees (1971), Atkinson and Stiglitz (1976), Akerlof (1978), Saez and Stantcheva (2016) Auctions with budget-constrained bidders: Che and Gale (1998), Fernandez and Gali (1999); Che, Gale, and Kim (2012); Pai and Vohra (2014); Kotowski (2017) Costly-screening/ money-burning literature: McAfee and McMillan (1992), Damiano and Li (2007), Hartline and Roughgarden (2008), Hoppe, Moldovanu, and Sela (2009), Condorelli (2012), Chakravarty and Kaplan (2013) Methods (generalized ironing): Muir and Loertscher (2021), Ashlagi, Monachou, and Nikzad (2021), Kleiner, Moldovanu, and Strack (2021), Z. Kang (2020a). New threads in mechanism design with redistribution motives: Z. Kang (2020b), M. Kang and Zheng (2020, 2021), Reuter and Groh (2021)

32. Model Model

33. Model A designer chooses a mechanism to allocate a unit mass of objects to a unit mass of agents. Each object has quality q ∈ [0, 1], q is distributed acc. to cdf F. Each agent is characterized by a type (i, r, λ), where: i is an observable label, i ∈ I (finite); r is an unobserved willingness to pay (for quality), r ∈ R+; λ is an unobserved social welfare weight, λ ∈ R+; The type distribution is known to the designer. If (i, r, λ) gets a good with quality q and pays t, her utility is qr − t, while her contribution to social welfare is λ(qr − t)

34. Model Comments for the energy context: Quality normalized to 0 or 1, interpreted as quantity. Then, q = 1 can be interpreted as the “normal consumption level.” Instead of distribution of quality F, we just have total quantity S available: F(q) = 1{q≥S}. Throughout, assume that S < 1. To the extent that some households have a higher “normal consumption level,” we can (intuitively) fold it into the WTP r r · qnormal = rqnormal | {z } r̃ · 1

35. Model The designer has access to arbitrary (direct) allocation mechanisms (Γ, T) where Γ(q|i, r, λ) is the probability that (i, r, λ) gets a good with quality q or less, and T(i, r, λ) is the associated payment, subject to: Feasibility: E(i,r,λ) [Γ(q|i, r, λ)] ≥ F(q), ∀q ∈ [0, 1]; IC constraint: Each agent (i, r, λ) reports (r, λ) truthfully; IR constraint: U(i, r, λ) ≡ r R qdΓ(q|i, r, λ) − T(i, r, λ) ≥ 0; Non-negative transfers: T(i, r, λ) ≥ 0, ∀(i, r, λ). The designer maximizes, for some constant α ≥ 0, a weighted sum of revenue and agents’ utilities: E(i,r,λ) [αT(i, r, λ) + λU(i, r, λ)] .

36. Comments about the model Lemma: It is without loss of optimality for the mechanism designer to only elicit information about r through the mechanism—allocation and transfers depend on (i, r) but not on λ directly.

37. Comments about the model Lemma: It is without loss of optimality for the mechanism designer to only elicit information about r through the mechanism—allocation and transfers depend on (i, r) but not on λ directly. This implies that the designer forms expectations over the unobserved social welfare weights: λi(r) ≡ E(i,r,λ) [λ| i, r] ; we call these functions Pareto weights, and assume they are continuous.

38. Comments about the model Lemma: It is without loss of optimality for the mechanism designer to only elicit information about r through the mechanism—allocation and transfers depend on (i, r) but not on λ directly. This implies that the designer forms expectations over the unobserved social welfare weights: λi(r) ≡ E(i,r,λ) [λ| i, r] ; we call these functions Pareto weights, and assume they are continuous. Designer’s objective becomes: E(i,r) [αTi(r) + λi(r)Ui(r)] .

39. Comments about the model λi(r) ≡ E(i,r,λ) [λ| i, r] Economic idea: The designer assesses the “need” of agents by estimating the unobserved welfare weights based on the observable (label i) and elicitable (wtp r) information.

40. Comments about the model λi(r) ≡ E(i,r,λ) [λ| i, r] Economic idea: The designer assesses the “need” of agents by estimating the unobserved welfare weights based on the observable (label i) and elicitable (wtp r) information. Consequence: The optimal allocation depends on the statistical correlation of labels and willingness to pay with the unobserved social welfare weights.

41. Comments about the model Correlation of labels with welfare weights: If label i captures income brackets, then we may naturally think that E[λ| i] ≡ λ̄i ≥ λ̄j ≡ E[λ| j] if i corresponds to a lower income bracket than j;

42. Comments about the model Correlation of labels with welfare weights: If label i captures income brackets, then we may naturally think that E[λ| i] ≡ λ̄i ≥ λ̄j ≡ E[λ| j] if i corresponds to a lower income bracket than j; Other possibilities (varies by country): Labels could capture family size, energy efficiency of housing, heating method, exposure to interest rate hikes.

43. Comments about the model Correlation of WTP with welfare weights: Suppose there is just one label I = {i}, but we elicit information about willingness to pay of two patients for a treatment: Agent A: e50,000 Agent B: e1,000 Naturally, we may presume that λi(r) is decreasing in r.

44. Comments about the model Correlation of WTP with welfare weights: Many labels, i captures low income, and we elicit information about willingness to pay of two patients for a treatment: Agent A: e 50,000 Agent B: e1,000 Naturally, we may presume that λi(r) is (“less”) decreasing in r.

45. Comments about the model Correlation of WTP with welfare weights: Suppose there is just one label I = {i}, but we elicit information about WTP of two consumers for maintaining “normal” energy consumption: Agent A: $10,000 Agent B: $5,000 Naturally, we may presume that λi(r) is decreasing in r.

46. Comments about the model Correlation of WTP with welfare weights: Many labels, i captures family size and energy efficiency, but we elicit information about WTP of two consumers for maintaining “normal” energy consumption: Agent A: $10,000 Agent B: $5,000 Naturally, we may presume that λi(r) is (“more”) decreasing in r.

47. Comments about the model The role of revenue: The weight α on revenue captures the value of a dollar in the designer’s budget, spent on the most valuable “cause.”

48. Comments about the model The role of revenue: The weight α on revenue captures the value of a dollar in the designer’s budget, spent on the most valuable “cause.” When α = maxi λ̄i, then it is as if a lump-sum transfer to group i were allowed; when α = averagei λ̄i, then it is as if lump-sum transfers to all agents were allowed.

49. Comments about the model The role of revenue: The weight α on revenue captures the value of a dollar in the designer’s budget, spent on the most valuable “cause.” When α = maxi λ̄i, then it is as if a lump-sum transfer to group i were allowed; when α = averagei λ̄i, then it is as if lump-sum transfers to all agents were allowed. When α > λ̄i for all i, there is an “outside cause”.

50. Comments about the model The role of revenue: The weight α on revenue captures the value of a dollar in the designer’s budget, spent on the most valuable “cause.” When α = maxi λ̄i, then it is as if a lump-sum transfer to group i were allowed; when α = averagei λ̄i, then it is as if lump-sum transfers to all agents were allowed. When α > λ̄i for all i, there is an “outside cause”. When α < λ̄i for some i, lump-sum payments to agents in group i are prohibited or costly (insurance motive as in Gadenne et al., 2022).

51. Comments about the model The role of revenue: The weight α on revenue captures the value of a dollar in the designer’s budget, spent on the most valuable “cause.” When α = maxi λ̄i, then it is as if a lump-sum transfer to group i were allowed; when α = averagei λ̄i, then it is as if lump-sum transfers to all agents were allowed. When α > λ̄i for all i, there is an “outside cause”. When α < λ̄i for some i, lump-sum payments to agents in group i are prohibited or costly (insurance motive as in Gadenne et al., 2022). When α = 0, we are in the money-burning/costly-screening case.

52. Comments about the model Universally desired goods

53. Comments about the model Universally desired goods Let Gi(r) be the cdf of willingness to pay conditional on label i, and let gi(r) be its continuous positive density on [ri, r̄i].

54. Comments about the model Universally desired goods Let Gi(r) be the cdf of willingness to pay conditional on label i, and let gi(r) be its continuous positive density on [ri, r̄i]. We call the good universally desired (for group i) if ri > 0.

55. Comments about the model Universally desired goods Let Gi(r) be the cdf of willingness to pay conditional on label i, and let gi(r) be its continuous positive density on [ri, r̄i]. We call the good universally desired (for group i) if ri > 0. Interpretation: A vast majority of agents have a willingness to pay that is bounded away from zero.

56. Comments about the model Universally desired goods Let Gi(r) be the cdf of willingness to pay conditional on label i, and let gi(r) be its continuous positive density on [ri, r̄i]. We call the good universally desired (for group i) if ri > 0. Interpretation: A vast majority of agents have a willingness to pay that is bounded away from zero. Examples: “Essential” goods: housing, energy, food, basic health care;

57. Comments about the model Universally desired goods Let Gi(r) be the cdf of willingness to pay conditional on label i, and let gi(r) be its continuous positive density on [ri, r̄i]. We call the good universally desired (for group i) if ri > 0. Interpretation: A vast majority of agents have a willingness to pay that is bounded away from zero. Examples: “Essential” goods: housing, energy, food, basic health care; “Stores of value”: shares, currencies, commodities

58. Derivation of Optimal Mechanism Derivation of Optimal Mechanism

59. Derivation of Optimal Mechanism 1 Objects are allocated “across” groups: F is split into I cdfs F⋆ i ;

60. Derivation of Optimal Mechanism 1 Objects are allocated “across” groups: F is split into I cdfs F⋆ i ; 2 Objects are allocated “within” groups: For each label i, the objects F⋆ i are allocated optimally to agents in group i. Details

61. Derivation of Optimal Mechanism 1 Objects are allocated “across” groups: F is split into I cdfs F⋆ i ; 2 Objects are allocated “within” groups: For each label i, the objects F⋆ i are allocated optimally to agents in group i. Details

62. Derivation of Optimal Mechanism 1 Objects are allocated “across” groups: F is split into I cdfs F⋆ i ; 2 Objects are allocated “within” groups: For each label i, the objects F⋆ i are allocated optimally to agents in group i. Details Observation: Only expected quality, Qi(r), matters for payoffs.

63. Derivation of Optimal Mechanism 1 Objects are allocated “across” groups: F is split into I cdfs F⋆ i ; 2 Objects are allocated “within” groups: For each label i, the objects F⋆ i are allocated optimally to agents in group i. Details Observation: Only expected quality, Qi(r), matters for payoffs. Result: Within a group, agents are partitioned into intervals according to WTP, with either the “market” or “non-market” allocation in each interval

64. Derivation of Optimal Mechanism 1 Objects are allocated “across” groups: F is split into I cdfs F⋆ i ; 2 Objects are allocated “within” groups: For each label i, the objects F⋆ i are allocated optimally to agents in group i. Details Observation: Only expected quality, Qi(r), matters for payoffs. Result: Within a group, agents are partitioned into intervals according to WTP, with either the “market” or “non-market” allocation in each interval Market allocation: assortative matching between WTP and quality Qi(r) = (F⋆ i )−1 (Gi(r)), ∀r ∈ [a, b];

65. Derivation of Optimal Mechanism 1 Objects are allocated “across” groups: F is split into I cdfs F⋆ i ; 2 Objects are allocated “within” groups: For each label i, the objects F⋆ i are allocated optimally to agents in group i. Details Observation: Only expected quality, Qi(r), matters for payoffs. Result: Within a group, agents are partitioned into intervals according to WTP, with either the “market” or “non-market” allocation in each interval Market allocation: assortative matching between WTP and quality Qi(r) = (F⋆ i )−1 (Gi(r)), ∀r ∈ [a, b]; Non-market allocation: random matching (rationing!) between WTP and quality Qi(r) = q̄, ∀r ∈ [a, b].

66. Derivation of Optimal Mechanism: within-group allocation

73. Derivation of Optimal Mechanism: across-group allocation

93. Economic Implications Economic Implications

94. Economic Implications Proposition 1 (WTP-revealed inequality ) Suppose that α ≥ λ̄i (lump-sum payments are available).

95. Economic Implications Proposition 1 (WTP-revealed inequality ) Suppose that α ≥ λ̄i (lump-sum payments are available). Then, it is optimal to ration agents with willingness to pay in some (non-degenerate) interval if and only if the function αJi(r) + Λi(r)hi(r) is not non-decreasing, where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r].

96. Economic Implications Proposition 1 (WTP-revealed inequality ) Suppose that α ≥ λ̄i (lump-sum payments are available). Then, it is optimal to ration agents with willingness to pay in some (non-degenerate) interval if and only if the function αJi(r) + Λi(r)hi(r) is not non-decreasing, where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r]. Assuming differentiability, the condition becomes Λ′ i(r)hi(r) + α + (Λi(r) − α)h′ i(r) < 0.

97. Economic Implications Proposition 1 (WTP-revealed inequality ) Suppose that α ≥ λ̄i (lump-sum payments are available). Then, it is optimal to ration agents with willingness to pay in some (non-degenerate) interval if and only if the function αJi(r) + Λi(r)hi(r) is not non-decreasing, where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r]. Assuming differentiability, the condition becomes Λ′ i(r)hi(r) + α + (Λi(r) − α)h′ i(r) < 0. Some non-market allocation is optimal if willingness to pay is strongly (negatively) correlated with the unobserved social welfare weights.

98. Economic Implications Proposition 2 (Optimality of uniform rationing) A necessary condition for uniform rationing (at a fixed low price) to be optimal within group i is that αr̄i ≤ Z r̄i ri rλi(r)dGi(r). This condition becomes sufficient if αJi(r) + Λi(r)hi(r) is quasi-convex.

99. Economic Implications Proposition 2 (Optimality of uniform rationing) A necessary condition for uniform rationing (at a fixed low price) to be optimal within group i is that αr̄i ≤ Z r̄i ri rλi(r)dGi(r). This condition becomes sufficient if αJi(r) + Λi(r)hi(r) is quasi-convex. Key take-away: It is never optimal to use uniform rationing if a lump-sum payment to group i is feasible.

100. Economic Implications Proposition 2 (Optimality of uniform rationing) A necessary condition for uniform rationing (at a fixed low price) to be optimal within group i is that αr̄i ≤ Z r̄i ri rλi(r)dGi(r). This condition becomes sufficient if αJi(r) + Λi(r)hi(r) is quasi-convex. Key take-away: It is never optimal to use uniform rationing if a lump-sum payment to group i is feasible. By an extension of the same argument: It is never optimal to have a uniform price cap.

101. Economic Implications Key idea behind optimality of rationing: Offering a reduced-price option with rationing allows to screen for vulnerable households.

102. Economic Implications Key idea behind optimality of rationing: Offering a reduced-price option with rationing allows to screen for vulnerable households. Each household chooses between a high per-unit price with no limit on consumption or a lower per-unit price but up to a consumption threshold q̄ < 1.

103. Economic Implications Key idea behind optimality of rationing: Offering a reduced-price option with rationing allows to screen for vulnerable households. Each household chooses between a high per-unit price with no limit on consumption or a lower per-unit price but up to a consumption threshold q̄ < 1. Selection will be based on WTP.

104. Economic Implications Key idea behind optimality of rationing: Offering a reduced-price option with rationing allows to screen for vulnerable households. Each household chooses between a high per-unit price with no limit on consumption or a lower per-unit price but up to a consumption threshold q̄ < 1. Selection will be based on WTP. Key question: Do we get the “right” selection?

105. Economic Implications Key idea behind optimality of rationing: Offering a reduced-price option with rationing allows to screen for vulnerable households. Each household chooses between a high per-unit price with no limit on consumption or a lower per-unit price but up to a consumption threshold q̄ < 1. Selection will be based on WTP. Key question: Do we get the “right” selection? If yes, subsidize (by reducing the per-unit price) the group that self-identified as being in need.

106. Economic Implications Key idea behind optimality of rationing: Offering a reduced-price option with rationing allows to screen for vulnerable households. Each household chooses between a high per-unit price with no limit on consumption or a lower per-unit price but up to a consumption threshold q̄ < 1. Selection will be based on WTP. Key question: Do we get the “right” selection? If yes, subsidize (by reducing the per-unit price) the group that self-identified as being in need. If no, then the rationing policy is pure allocative inefficiency.

107. Economic Implications Key idea behind optimality of rationing: The answer to the question largely depends on available labels!

108. Economic Implications Key idea behind optimality of rationing: The answer to the question largely depends on available labels! Mathematically, we need strong negative correlation between WTP and welfare weights (Λ′ i(r) < 0)

109. Economic Implications Key idea behind optimality of rationing: The answer to the question largely depends on available labels! Mathematically, we need strong negative correlation between WTP and welfare weights (Λ′ i(r) < 0) This might hold when we control for confounding factors that wold raise the WTP of poor households:

110. Economic Implications Key idea behind optimality of rationing: The answer to the question largely depends on available labels! Mathematically, we need strong negative correlation between WTP and welfare weights (Λ′ i(r) < 0) This might hold when we control for confounding factors that wold raise the WTP of poor households: ▶ Large family size;

111. Economic Implications Key idea behind optimality of rationing: The answer to the question largely depends on available labels! Mathematically, we need strong negative correlation between WTP and welfare weights (Λ′ i(r) < 0) This might hold when we control for confounding factors that wold raise the WTP of poor households: ▶ Large family size; ▶ Energy inefficiency of housing;

112. Economic Implications Key idea behind optimality of rationing: The answer to the question largely depends on available labels! Mathematically, we need strong negative correlation between WTP and welfare weights (Λ′ i(r) < 0) This might hold when we control for confounding factors that wold raise the WTP of poor households: ▶ Large family size; ▶ Energy inefficiency of housing; ▶ Electricity-based heating.

113. Economic Implications Key idea behind optimality of rationing: Advantage of consumption-capped price subsidy over lump-sum payments: Additional screening!

114. Economic Implications Key idea behind optimality of rationing: Advantage of consumption-capped price subsidy over lump-sum payments: Additional screening! Market choices may reveal which households are really in need.

115. Economic Implications Key idea behind optimality of rationing: Advantage of consumption-capped price subsidy over lump-sum payments: Additional screening! Market choices may reveal which households are really in need. Then, clear why uniform price and quantity caps can’t work!

116. Economic Implications Key idea behind optimality of rationing: Advantage of consumption-capped price subsidy over lump-sum payments: Additional screening! Market choices may reveal which households are really in need. Then, clear why uniform price and quantity caps can’t work! They distort economic efficiency without offering the screening benefit.

117. Economic Implications Key idea behind optimality of rationing: Advantage of consumption-capped price subsidy over lump-sum payments: Additional screening! Market choices may reveal which households are really in need. Then, clear why uniform price and quantity caps can’t work! They distort economic efficiency without offering the screening benefit. Thus, they are dominated by lump-sum payments (here, we agree with the Economist and the IMF paper)

118. Economic Implications Key idea behind optimality of rationing: Advantage of consumption-capped price subsidy over lump-sum payments: Additional screening! Market choices may reveal which households are really in need. Then, clear why uniform price and quantity caps can’t work! They distort economic efficiency without offering the screening benefit. Thus, they are dominated by lump-sum payments (here, we agree with the Economist and the IMF paper) What if lump-sum payments are costly? (administration costs, insurance motive, behavioral aspects, etc.)

119. Economic Implications Proposition 3 (Label-revealed inequality ) Suppose that the average Pareto weight λ̄i for group i is strictly larger than the weight on revenue α; the good is universally desired (ri > 0). Then, there exists r⋆ i > ri such that the optimal allocation is to ration (at a price of 0) all types r ≤ r⋆ i .

120. Economic Implications Proposition 3 (Label-revealed inequality ) Suppose that the average Pareto weight λ̄i for group i is strictly larger than the weight on revenue α; the good is universally desired (ri > 0). Then, there exists r⋆ i > ri such that the optimal allocation is to ration (at a price of 0) all types r ≤ r⋆ i . Interpretation: If the designer would like to redistribute to group i but cannot give agents in that group a direct cash transfer, then it is optimal to provide small quantities of universally desired goods for free.

121. Economic Implications Proposition 3 (Label-revealed inequality ) Suppose that the average Pareto weight λ̄i for group i is strictly larger than the weight on revenue α; the good is universally desired (ri > 0). Then, there exists r⋆ i > ri such that the optimal allocation is to ration (at a price of 0) all types r ≤ r⋆ i . Interpretation: If the designer would like to redistribute to group i but cannot give agents in that group a direct cash transfer, then it is optimal to provide small quantities of universally desired goods for free. Note: There might still be a price gradient for higher quantities (the exact condition is the one we have seen before)

122. Economic Implications Proposition 1 (Label-revealed inequality ) Suppose that the average Pareto weight λ̄i for group i is strictly larger than the weight on revenue α; the good is universally desired (ri > 0). Then, there exists r⋆ i > ri such that the optimal allocation is to ration (at a price of 0) all types r ≤ r⋆ i . “Wrong” intuition: A random allocation for free increases the welfare of agents with lowest willingness to pay

123. Economic Implications Proposition 1 (Label-revealed inequality ) Suppose that the average Pareto weight λ̄i for group i is strictly larger than the weight on revenue α; the good is universally desired (ri > 0). Then, there exists r⋆ i > ri such that the optimal allocation is to ration (at a price of 0) all types r ≤ r⋆ i . “Wrong” intuition: A random allocation for free increases the welfare of agents with lowest willingness to pay Correct Intuition: A random allocation for free enables the designer to lower prices for all agents Picture

124. Conclusions Conclusions

125. Conclusions When to use in-kind redistribution? (provision of goods at a below-market-clearing prices)

126. Conclusions When to use in-kind redistribution? (provision of goods at a below-market-clearing prices) Roughly: When observable variables uncover inequality in the unobserved welfare weights.

127. Conclusions When to use in-kind redistribution? (provision of goods at a below-market-clearing prices) Roughly: When observable variables uncover inequality in the unobserved welfare weights. 1 Label-revealed inequality:

128. Conclusions When to use in-kind redistribution? (provision of goods at a below-market-clearing prices) Roughly: When observable variables uncover inequality in the unobserved welfare weights. 1 Label-revealed inequality: When lump-sum payments are costly and some groups can be identified as vulnerable, it is optimal to use some in-kind redistribution for universally desired goods.

129. Conclusions When to use in-kind redistribution? (provision of goods at a below-market-clearing prices) Roughly: When observable variables uncover inequality in the unobserved welfare weights. 1 Label-revealed inequality: When lump-sum payments are costly and some groups can be identified as vulnerable, it is optimal to use some in-kind redistribution for universally desired goods. ▶ Energy context: Controlling electricity bills for poorest households may be easier/ cheaper than direct cash transfers.

130. Conclusions When to use in-kind redistribution? (provision of goods at a below-market-clearing prices) Roughly: When observable variables uncover inequality in the unobserved welfare weights. 1 Label-revealed inequality: When lump-sum payments are costly and some groups can be identified as vulnerable, it is optimal to use some in-kind redistribution for universally desired goods. ▶ Energy context: Controlling electricity bills for poorest households may be easier/ cheaper than direct cash transfers. 2 WTP-revealed inequality: When welfare weights are strongly and negatively correlated with willingness to pay conditional on labels.

131. Conclusions When to use in-kind redistribution? (provision of goods at a below-market-clearing prices) Roughly: When observable variables uncover inequality in the unobserved welfare weights. 1 Label-revealed inequality: When lump-sum payments are costly and some groups can be identified as vulnerable, it is optimal to use some in-kind redistribution for universally desired goods. ▶ Energy context: Controlling electricity bills for poorest households may be easier/ cheaper than direct cash transfers. 2 WTP-revealed inequality: When welfare weights are strongly and negatively correlated with willingness to pay conditional on labels. ▶ Energy context: A well-designed subsidy system can screen for most vulnerable households, better than a cash transfer scheme could.

132. Conclusions We proposed a model of Inequality-aware Market Design.

133. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors:

134. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors: 1 Correlation between the unobserved welfare weights and the information that the designer can elicit or observe directly;

135. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors: 1 Correlation between the unobserved welfare weights and the information that the designer can elicit or observe directly; 2 Importance of availability of lump-sum transfers;

136. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors: 1 Correlation between the unobserved welfare weights and the information that the designer can elicit or observe directly; 2 Importance of availability of lump-sum transfers; 3 Whether the good is universally desired or not.

137. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors: 1 Correlation between the unobserved welfare weights and the information that the designer can elicit or observe directly; 2 Importance of availability of lump-sum transfers; 3 Whether the good is universally desired or not. More work is needed on market design with redistributive concerns:

138. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors: 1 Correlation between the unobserved welfare weights and the information that the designer can elicit or observe directly; 2 Importance of availability of lump-sum transfers; 3 Whether the good is universally desired or not. More work is needed on market design with redistributive concerns: 1 There are hundreds (thousands?) of theory papers focusing on how to maximize revenue or efficiency when allocating resources...

139. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors: 1 Correlation between the unobserved welfare weights and the information that the designer can elicit or observe directly; 2 Importance of availability of lump-sum transfers; 3 Whether the good is universally desired or not. More work is needed on market design with redistributive concerns: 1 There are hundreds (thousands?) of theory papers focusing on how to maximize revenue or efficiency when allocating resources... 2 ... but only a handful on how to achieve optimal redistribution.

140. Conclusions We proposed a model of Inequality-aware Market Design. Our analysis uncovers the importance of three factors: 1 Correlation between the unobserved welfare weights and the information that the designer can elicit or observe directly; 2 Importance of availability of lump-sum transfers; 3 Whether the good is universally desired or not. More work is needed on market design with redistributive concerns: 1 There are hundreds (thousands?) of theory papers focusing on how to maximize revenue or efficiency when allocating resources... 2 ... but only a handful on how to achieve optimal redistribution. 3 Much more work remains to be done... (many stylized assumptions in our model!)

141. Thank you Thank you

142. Appendix Appendix

143. Back to slides

144. Back to slides

145. Back to slides

146. Back to slides

147. Back to slides

148. Derivation of the optimal “within-group” mechanism The “within-group” problem: Fixing a group of agents i, and a distribution of quality Fi available for group i, what is the optimal way to allocate quality subject to IC and IR constraints?

149. Derivation of the optimal “within-group” mechanism The designer’s problem: (Qi(r) denotes expected quality) max Qi(r), Ui(ri) Z r̄i ri Vi(r)Qi(r)dGi(r) + (λ̄i − α)Ui(ri)

150. Derivation of the optimal “within-group” mechanism The designer’s problem: (Qi(r) denotes expected quality) max Qi(r) Z r̄i ri Vi(r)Qi(r)dGi(r) + max{0, λ̄i − α}riQi(ri).

151. Derivation of the optimal “within-group” mechanism The designer’s problem: (Qi(r) denotes expected quality) max Qi(r) Z r̄i ri Vi(r)Qi(r)dGi(r) + max{0, λ̄i − α}riQi(ri). In the above, Vi(r) = αJi(r) + Λi(r)hi(r), where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r].

152. Derivation of the optimal “within-group” mechanism The designer’s problem: (Qi(r) denotes expected quality) max Qi(r) Z r̄i ri Vi(r)Qi(r)dGi(r) + max{0, λ̄i − α}riQi(ri). In the above, Vi(r) = αJi(r) + Λi(r)hi(r), where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r]. Feasible distribution of expected quality: CDF Φi such that Fi ≻MPS Φi.

153. Derivation of the optimal “within-group” mechanism The designer’s problem: (Qi(r) denotes expected quality) max Qi(r) Z r̄i ri Vi(r)Qi(r)dGi(r) + max{0, λ̄i − α}riQi(ri). In the above, Vi(r) = αJi(r) + Λi(r)hi(r), where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r]. Feasible distribution of expected quality: CDF Φi such that Fi ≻MPS Φi. Incentive-compatibility =⇒ Gi(r) = Φi(q)

154. Derivation of the optimal “within-group” mechanism The designer’s problem: (Qi(r) denotes expected quality) max Qi(r) Z r̄i ri Vi(r)Qi(r)dGi(r) + max{0, λ̄i − α}riQi(ri). In the above, Vi(r) = αJi(r) + Λi(r)hi(r), where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r]. Feasible distribution of expected quality: CDF Φi such that Fi ≻MPS Φi. Incentive-compatibility =⇒ Qi(r) = Φ−1 i (Gi(r))

155. Derivation of the optimal “within-group” mechanism The designer’s problem: (Qi(r) denotes expected quality) max Qi(r) Z r̄i ri Vi(r)Qi(r)dGi(r) + max{0, λ̄i − α}riQi(ri). In the above, Vi(r) = αJi(r) + Λi(r)hi(r), where hi(r) = 1 − Gi(r) gi(r) , Ji(r) = r − hi(r), Λi(r) = Ei [λi(r̃) | r̃ ≥ r]. Feasible distribution of expected quality: CDF Φi such that Φ−1 i ≻MPS F−1 i . Incentive-compatibility =⇒ Qi(r) = Φ−1 i (Gi(r))

156. Derivation of the optimal “within-group” mechanism The designer’s problem is max Ψi Z r̄i ri Vi(r)Ψi(Gi(r))dGi(r) + max{0, λ̄i − α}riΨi(0) s.t. Ψi ≻MPS F−1 i Back

157. Derivation of the optimal “within-group” mechanism The designer’s problem is max Ψi Z 1 0 Vi(t) z }| { Z 1 t Vi(G−1 i (x))dx + max{0, λ̄i − α}ri1{t=0} dΨi(t) s.t. Ψi ≻MPS F−1 i Back

158. Derivation of the optimal “within-group” mechanism The designer’s problem is max Ψi Z 1 0 Vi(t) z }| { Z 1 t Vi(G−1 i (x))dx + max{0, λ̄i − α}ri1{t=0} dΨi(t) s.t. Ψi ≻MPS F−1 i Agents are partitioned into intervals according to WTP: Back

159. Derivation of the optimal “within-group” mechanism The designer’s problem is max Ψi Z 1 0 Vi(t) z }| { Z 1 t Vi(G−1 i (x))dx + max{0, λ̄i − α}ri1{t=0} dΨi(t) s.t. Ψi ≻MPS F−1 i Agents are partitioned into intervals according to WTP: ▶ Market allocation: assortative matching between WTP and quality Qi(r) = F−1 i (Gi(r)), ∀r ∈ [a, b]; Back

160. Derivation of the optimal “within-group” mechanism The designer’s problem is max Ψi Z 1 0 Vi(t) z }| { Z 1 t Vi(G−1 i (x))dx + max{0, λ̄i − α}ri1{t=0} dΨi(t) s.t. Ψi ≻MPS F−1 i Agents are partitioned into intervals according to WTP: ▶ Market allocation: assortative matching between WTP and quality Qi(r) = F−1 i (Gi(r)), ∀r ∈ [a, b]; ▶ Non-market allocation: random matching between WTP and quality Qi(r) = q̄, ∀r ∈ [a, b]. Back

161. Derivation of the optimal “within-group” mechanism The designer’s problem is max Ψi, F̃i Z 1 0 Vi(t) z }| { Z 1 t Vi(G−1 i (x))dx + max{0, λ̄i − α}ri1{t=0} dΨi(t) s.t. Ψi ≻MPS F̃−1 i ≻FOSD F−1 i Agents are partitioned into intervals according to WTP: ▶ Market allocation: assortative matching between WTP and quality Qi(r) = F−1 i (Gi(r)), ∀r ∈ [a, b]; ▶ Non-market allocation: random matching between WTP and quality Qi(r) = q̄, ∀r ∈ [a, b]. Back

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